Author: tombennison

The Japanese method of multiplication seems to be everywhere at the moment – Julia (@tessmaths) I noticed had advocated it ( I think prompted by a tweet by @missradders) on Twitter overtone weekend and one of my oldest friends posted a video of it on my Facebook page yesterday asking how it works and if it will always work. 

This method is apparently taught to Japanese primary school pupils (note to self: ask the Japanese exchange pupils when they next come!) as an easy method for multiplying large numbers (larger than the times tables anyway). 

It works due to the way that numbers are written down in base 10. For example, 325 is 3 lots of a hundred plus 20 lots of ten plus 5 lots of one. This along with the distributive property of multiplication allows us to split numbers up when multiplying. If I was working out 24 times 12 mentally I would split it up into two multiplications – 20 times 12 and 4 times 12 – then add the results. Mathematically, this can be written as:

\( 24 \times 12 = (20 + 4) \times 12 = 240 + 48 = 288 \)

The Japanese method takes this a step further and says that both numbers can be split up in this way and so that 

\(\) 24 \times 12 = (20 + 4) \times (10+2) = 20 \times 10 + 20 \times 2 + 4 \times 10 + 4 \times 2 = 200 + 40 + 40 + 8 = 288 \(\) 

To multiply using the Japanese method you represent 24 as two parallel lines, a large gap and then another parallel line and represent 12 as 1 parallel line with a gap then 2 further parallel lines. The lines for 24 and 12 cross each other. Then to calculate the product you count the intersections on the right for the units column, the tens is calculated by combining the two sets of intersections in the middle and then the number in the hundreds column from counting the intersections on the left. 

I think that makes a lot more sense in a picture, so here is another example:

  
Some are harder than others:  

 

In the example above the number of intersections in  the middle is 16 and so 10 of them have to be carried to the left, increasing the number of hundreds from 6 to 7.

For larger numbers this method becomes incredibly cumbersome – what withdrawing all the lines, counting intersections and dealing with the carries. See the example below for two 3 digit numbers: 

 I much prefer the lattice/Chinese/Napiers Bones method myself, but that is for another day. 

Firstly, after a discussion on Twitter the other week I’ve decided to delay publishing my workings of the Edexcel exam papers until I know that they are already out there on the Internet. My personal feeling is that the exam papers (not solutions) should be made publicly available (I.e. Not in any secure sections of websites) by the exam boards no more than a week after they have been sat. I can’t see how there is any commercial value in them not doing this, and these days a past paper can’t be used as a Mock (not least because they are released by the exam boards before mocks are likely to take place for A Levels) without students having the potential to have seen them. Having said that, these posts will become password protected before the next academic year – any teachers who want a password just email me. 

Anyway, back to the topic of this post – the Edexcel Mechanics 2 exam; my students were pretty worried about this paper. In light of this I think that the paper was pretty nice – not quite as hard as in previous years, but still some things to challenge the students, especially those who don’t like non-standard questions. 

The paper started nicely, with Question 1 being a fairly straightforward  application of F=ma in conjunction with work done. 

 

I really liked Question 2! The first part is a standard centres of mass of a lamina, but the second part is trickier and isn’t a “standard” question so to speak. As long as you apply the knowledge of what it means for the lamina with the extra weight hanging vertically through a particular line then, with the help of a good diagram, finding the extra mass is a nice problem to solve. 

   

Question 3  is a simple exercise in applying the definition of impulse and then finding out the increase in kinetic energy following the impulse. 

 

Stuart ( @sxpmaths) commented on Twitter that Question 4 is a classic example of the type of question where all intermediate steps are removed (I talk about these kind of questions here). As long as the students are ok at drawing the diagram then the question isn’t any harder than if the diagram had been given. The difficulty is increased though as the question doesn’t guide you through the necessary steps.   
 For Question 5, you’ll see that I made some silly calculation errors in working out the solution to part b – my excuse: it was late!!     

 Question 6 was a (in my opinion) a standard kinematics with variable acceleration question – nothing non standard about this.   The projectiles question was non-standard in the sense that you don’t often see questions with the angle of the particle at a particular (non-initial) point in the motion. Nice numbers hadn’t been used either so you ended up with horrible decimal answers – hopefully students stored the full values in their calculator for subsequent parts. I actually think this should be a requirement, and if they don’t then at most 1/2 marks for subsequent parts – a bit harsh I know. 

  

   

   

The final question  was a really nice projectiles and conservation of linear momentum question finding an inequality for the coefficient of restitution and then investigating further questions. 

  

  

  Overall I think that like most Edexcel papers this year, this paper wasn’t really nice for the students but it could definitely have been a lot harder! 

Whilst I was out tutoring last night I missed a post from Jo (@mathsjem ) about Prisms… There was a debate about whether a cylinder was a prism.

I had always understood a requirment of a solid being a prism was that the base of a prism was polygonal – i.e. made up of straight lines. This means that a cylinder cannot be a prism. However, the formulae sheets from GCSE papers seem to disagree, as this picture from the Edexcel papers shows:

Here we have a “prism” that clearly has curved edges forming the cross section.

I know quite often the volume of a cylinder is taught by referring to it as a prism with a circular cross section, but I haven’t seen any reliable definitions of prism that include cross sections with curved boundary. Instead, all definitions I have seen specifically require that the face is polygonal, for instance this definition from Wolfram’s Mathworld. Here they reference an old book on solid geometry as a source of this definition – Solid Mensuration with Proofs by Kern & Bland – @El_Timbre do you have a copy of this by any chance?

Keith (@MrKMorrison ) suggested that the word prism comes from the greek ‘prisma’ which literally means ‘something sawed’, suggesting the same face throughout, and so a cylinder should be a prism. I think the root of the word is largely immaterial, once an accepted definition is present. For instance multiply is a word in general usage and with a precise mathematical meaning, the root of multiply is the old French (I believe) for increase, but mathematically we wouldn’t use the word multiply to mean this. 

In the grand scheme of things I guess it isn’t really a big deal to call a cylinder a prism – a cylinder certainly “behaves” like a prism, and maybe for lower attaining students calling a cylinder a special prism perhaps helps. For students going on to study maths though I think being loose with definitions can lead to problems, especially once they get to university. 

I alluded to prime numbers in a tweet this morning as being what I think is the most dangerous example of miss-teaching of definitions / miss understanding of a definition. I’ve had so many students (including undergraduates) say to me that 1 is prime because it only has 1 and itself as factors. This is despite the definition of a prime number explicitly excluding 1. I’ve always found this quite odd……

Update: Mike Lawler (@mikeandallie) provides me a link to this nice definition from the Art of Problem Solving 

I’m not sure why (someone probably asked me at school) but yesterday I started thinking about what I think leads to a good ethos in an A Level class and what I strive to achieve with my classes. 

My guiding principle with A Level groups is equality – I don’t want to be seen as a teacher who is above them. Real mathematics is a collaborative attack on problems where you are looking for an elegant, neat solution; this is hard to achieve amongst a group of people who don’t feel equal. I don’t want them to see me as a teacher above them who they can only ask questions of – I want them to suggest approaches and paths to the solution of problems too. I think one of the easiest ways to encourage this is to let them call me by my first name. I know some teachers wouldn’t like this, and it is of course important to keep a professional distance from your students, but, I really think little things like this encourage them to be comfortable to give me their thoughts and suggest ways to tackle a question that they are doing as a class. 

Another thing that I think is important for an A Level group to understand is the necessity to be stuck. In a Further Maths group especially some of the students have probably never struggled with maths before and it can be a shock when they begin to. For this reason I try to emphasise in the first few weeks that this is ok and a normal part of doing mathematics. Problems which either can’t be solved, or can be solved in multiple approaches, one of which takes significantly longer than another approach are valuable to reinforce that being stuck is ok! I also use exercises that I haven’t looked at until I get into the room as examples for this reason – I believe it’s good for them to see that I don’t always do the correct thing first time when solving a problem. 

Collaborative working is something that I try to foster with certain activities – certainly for me, maths at A Level was a very solitary activity. This isn’t really reflective of the world of mathematics and whilst it is important for students to have plenty of  practice at solving questions on their own, talking about and discussing mathematics is incredibly valuable. 

This post is a work in progress and I’m going to add to it as I think of other things….. 

I don’t think anyone in the UK can have missed the media frenzy over the first Edexcel maths GCSE paper of this year. For example see here and here.

Having done the paper I don’t think it was particularly unfair or hard. There were plenty of standard, well trodden questions aimed at the C/B range. The questions targeting students looking for an A or A* are designed to stretch the most able and should be hard. I think there was a clear change in how se of these questions were written and I suspect this is a reflection of questions to come with the new GCSE. 

One of the questions that has gone viral is the “Hannah’s Sweets” question  

 

Apart from the pseudo-context (I don’t see how you could know this probability without knowing how many sweets we’re in the bag to begin with) I quite like this question. It brings together probability and algebra, using algebra as it was designed – a tool for solving problems. In fact, as long as a student writes out what they know from the question it is in fact fairly easy. At A Level I emphasise writing out what you know from a question if you don’t know where to start, but in the past I haven’t really done this to the same extent with my GCSE classes. 

This question, however, is unlike any of the probability questions on recent year’s past papers and this is, I think, where the problem lies. I try not to teach to the test, but the fact that some of my students struggled with this question shows (depressingly) that to some extent I do. A lot of my revision lessons, probably like most teachers, have focussed on past paper questions as a way to prepare students – for this question this approach has failed. 

From what I remember in one of the new specimen assessment materials there is a tough looking probability question involving a spinner similar to this… To prepare our students for these new exams I think I am going to need to change how I do exam preparation at GCSE. I don’t think this is necessarily a bad thing, maths is a problem solving tool, students shouldn’t just be expecting to see the same style questions year in, year out. I suspect that past papers for the new GCSE won’t be able to be used to coach pupils into the question style, at least not for the high attaining questions. 

This is a good thing in my opinion, but my main concern though is the time necessary to build this deep understanding, especially for the first couple of cohorts for the new papers who won’t have had the required preparation at Key Stage 3. For The new Year 7 we are going down the mastery route and this should allow us the time to build this deep conceptual grasp of topics and how they inter-relate which will be good. 

Going back to this year’s Paper 1 another question that a lot of my students were talking about was the “conical grain store” question. I loved this question, nice numbers to work with and it essentially boils down to a pair of simultaneous equations to solve using substitution. Substitution is a method that in hindsight, I strangely down do enough of at GCSE, but would use almost exclusively with an A Level group. The question concerning the perimeter of a shape made up of 4 congruent triangles and 4 congruent rectangles was also nice – just Pythagoras in disguise. I’d wager a bet though that a large portion of students didn’t even attempt these questions due to the unfamiliar context. 

All in all I think it was a completely fair paper, and I’m looking forward to seeing tomorrow mornings paper. I wonder if there will be another Twitter frenzy…. 

Update: Ed at Solvenymaths (@solvemymaths) has written a fantastic post with some thoughts on how to address the gap between current student’s problem solving skills and what will be required for the new style GCSE questions. 

Yesterday morning students at my school sat M1, FP2 and the OCR FMSQ papers. This is the first of three posts looking at these papers and I’ll start with the Mechanics paper. 

I’ll admit I’m feeling pretty tired at the moment and so I made a few mistakes that I had to correct as I went through the paper – have fun spotting them. Overall I think it was a standard M1 paper which included the obligatory question using a 3-4-5 triangle. 

Question 1 was a very straight forward conservation of momentum question – as long as you don’t get confused with the direction the particles are moving in there are easy marks to get here.  

 

I thought that the projectiles question on this paper (Question 2) was really nice; no projection at an angle, just straightforward motion under gravity. Nice numbers too so that the arithmetic was relatively easy. The last part of this question was straightforward, though I can see some people working out the time to get from 14.7m to 19.6m and then the time back down instead of using symmetry.  

 

My students commented on how Question 3 was nice as you got 7 marks for just resolving in two directions. A very simple question really as long as you are confident with solving simultaneous equations. 

 

The first question that I made an elementary mistake on was Question 4. This year ion concerned a lift inside of which was a crate. If you recognise when to consider the motion of the system and the crate separately it is pretty easy. However, when I district did it and considered the motion of the lift-crate system I included the normal reaction of the crate in my force balance – very silly of me.  

 

Question 5 was a typical moments question with a bar suspended from the ceiling and with particles suspended from it.  Taking moments about two points allows you to easily find the tension in the ropes for the first part. The second part is marginally trickier as you need to have an understanding of how the rod would move if the particle was changed. 

     

Question 6 part a seems hardly worth asking as a distinct question, but I guess it is only worth one mark. I have also realised I did part c in a much more complicated way than necessary 🙁 

 

I found question 7 strange as I realised that I had worked out most of part b as I drew the graph. So maybe it is only the shape of the graph and the relative gradients of the acceleration and deceleration that they are looking for in the mark scheme of the first part        . Finding V for the final part of the question is straight forward given the stepping stones you are led through in part b. 

 

Overall I liked the final question. However, I remember that when I did my A Levels finding the resultant force exerted by a string on the pulley was something that I struggled with. Indeed this is the part of the paper that most of my students said they struggled with. They also all said they were surprised that the first part of this question was worth 11 marks.  I agree I think, it could easily have been worth less marks so it’s quite a nice question to end the paper on I think 

   

I’d be interested to know what you or your students thought of the paper….

Yesterday I was pleased to notice that the packaging for the new recipe flavoured milks includes some nice maths puzzles on three of the four varieties (only the chocolate fudge flavour doesn’t contain a number Puzzle)  

   
 

I think these are pretty nice for their target audience – which I guess is a lot younger than me!! 

Go on, spend a minute and give them a go 😉

Next year I am likely to be teaching Mechanics 3 to my further mathematicians and so I thought I should have a go at this years exam paper (1 person sat it at school so I had a copy of it). After seeing this video I was a little nervous as I hadn’t looked at this since I did my A Levels, however I didn’t find it too bad with a bit of help from my old A Level textbook:

   
Question 1 I thought was pretty straightforward – once I had refreshed my memory of how to calculate elastic potential energy it was just an energy conservation question. Please excuse that I have written Hooke without a capital letter.

Question 2 was nice, as long as you could remember how to find volumes of revolution, and then use this to find \(\bar{x}\)

  

  

For Question 3 I realised that sometimes I have no intuition with the mechanics questions. Even though it was asking you to find the tensions in each string I still expected them to come out equally – of course in hindsight this clearly wouldn’t have made much sense. The circular motion stuff came back to me quicker than I expected to be honest, and this question dropped out quite nicely. 

   

Question 4 was a nice power type question, I thought it was very like an M2 question, just with the complication of non constant acceleration. I really like how the question required you to use the Trapezium rule. As a numerics guy I think how the numerical methods are presented at A Level is incredibly sad. The trapezium rule is great, and it could be used so much in the applied modules – students wouldn’t like being asked to use something from Core 2 in other modules though. I think I may have to write about the Trapezium rule……   

 

Question 5 is a nice centres of mass of a 3D solid. It considers a spindle formed of two cones. In hindsight it would have made more sense for me to work out the Center of mass using moments from A instead of taking moments from B

   

I can remember loving questions like Question 6 when I did A level, and I still quite enjoy showing that a particle connected to two springs exhibits simple harmonic motion. Like many questions as long as you are comfortable with applying F=ma and solving simultaneous equations it is fairly straight forward. 

     

The final question considers a particle moving on the surface of a sphere and uses conservation of energy and F=ma. I wouldn’t be surprised if some students forgot to add the horizontal distance moved whilst on the sphere in the final part.   

     

Formula triangles are quite prevalent in the UK (I’m not about elsewhere to be honest) – especially in science lessons and when right angled triangle trigonometry is taught. A typical example is the following for the relationship between speed, distance and time:

  
I’ve expressed my dislike (hatred is perhaps too strong a word) of formula triangles on Twitter before and others have written about them including Stephen Cavadino. Like Stephen my main reason for not being a fan of them is that they are usually introduced as a “trick” for use in particular situations with no reference to the underlying mathematics. 

One of my students when looking at rearranging formulae was asked to make t the subject of an equation and drew a formula triangle as shown below (I’ve re-written it!)  

When I asked why they had drawn a formula triangle they responded “well there are 3 terms so it’s a formula triangle question”. I’ve never seen this misconception before, I guess it is maybe down to formula triangles being used only for 3 term formulae and then this link becoming solidified in the student. 

When I probed a bit further and asked things like “why have you picked 4a to be on the top” after a while they realised their mistake. I can’t help thinking that this wouldn’t have come about if they had never seen a formula triangle and instead had just had plenty of opportunity to practice rearranging formulae. 

Yesterday I went to an Intermediate Geogebra course at the Geogebra Institute of Sheffield run by Mark Dabbs. It  was really good, and I have picked up lots of things I hadn’t realised (but probably should have done). For example, the fact that the input bar can be moved to the top of the screen so that students can more easily see it when projected onto a screen. Or that it is possible to adjust properties of many objects by just highlighting them.

I have lots of ideas of Geogebra things to do, which I’m sure I will post on here as I do them, but I thought I’d quickly share a trick (I’m sure some of you already do this) that gets rid of a little niggle of mine.

When teaching circle theorems I think it is nice to let them have a bit of time to discover them for themselves. With Geogebra you can easily create dynamic worksheets for them to explore. I’ve had a few sheets like the one below for demonstrating the “angle subtended at the centre” theorem

The angle at the centre is obviously meant to be twice the angle at the circumference but \( 2 \times 47.66 = 95.32 \) which is not \(95.31 \). Of course this is just an artefact of the rounding done to represent the angle to two decimal places, but it does distract from what I am hoping the students will spot. This could prompt a nice discussion about rounding errors and limits of accuracy, and of course could be mitigated somewhat by disiplaying more decimal places. But, I had always thought it would be nice if I could restrict the angle at the centre to prevent this from happening – which it turns out is easy to do. Using GeoGebra’s Sequence command (which behaves much like a for loop in a conventional programming language), we can generate a set of points around the circle such that the angle at the centre is either a whole number or a multiple of a half. This means that the angle at the edge will always be exactly represented in the two decimal places restriction.

Once the circle with centre A has been created (by default it is given the name c), I placed one point on the circumference, call it A’ say with the command
[code] A + (Radius[ c],0] [/code]. I then rotated this single point around the circle in increments of  0.5 degrees using the command [code] Sequence[Rotate[A’,kº,A],k,1,360,0.5] [/code]. This rotates the point A’ about the centre of the circle by k degrees in increments of 0.5 degrees, creating a list of points, which by default is named list1. Then  you need to hide the points around the circle from the graphics view, before creating three points B,C and D on the circumference. Once you have added in the appropriate line segments the angle can be added in using the angl tool. The final step is to redefine the definition of the points B, C and D from [code] Point[ c] [/code] to [code] Point[list1] [/code] as shown below in the screen shot from the iPad app – it is much easier to do this in the desktop version.

I have hosted both versions of these basic applets on my website here.

I really would recommnd that anyone who likes using GeoGebra to attend one of Mrk’s courses, it was a great way to spend a few hours and I am hoping to go to the advanced course in July.